I'm Busy

Recently, I'm a little bit busy.

I left my home town and went to Seoul^^

I am looking for a new house to stay.

Recently, I'm a little bit busy. (This is more natural, but what you said is fine.)

I leaved left my hometown (Don't put a space.) and I went to Seoul^^

I have to start everything again(I have to start all over again.)

I am finding (or "I am looking for") to stay my a new house to stay.

If it isn't exactly a "house", you can just say "a new place to stay".

Now, I'm having a very hard time but I will be fine^^ (Now, I'm having difficulty but I will be fine.)

I left my hometown, and I moved to Seoul^^

The other two covered everything else. Saying you "went" there can mean you went there to visit (which I thought at first) but since you are living there now, it is safe to put "I moved to Seoul."

I've Been Busy Lately

요즘 제가 좀 바빠요. = I've been a bit busy these days.

I've been (a little bit) busy these days. <

고향을 떠나 서울로 올라왔어요. = I left my hometown and came to Seoul.

I had to leave my hometown and move to Seoul.

서울에서 모든 것들을 다시 시작해야 해요. = I'll have to make a fresh start in Seoul.

I have to start all over here now.

제가 머물려야 할 새로운 집을 찾고 있어요 = I'm looking for a new home to stay in. (a new place to stay.)

I'm looking forward to living in my new home.

지금은 힘들지만 곧 괜찮아 질 거에요. = Things are very difficult right now, but it'll all be okay soon.

I'm having a hard time right now, but it'll be okay.